Linear magnetoelastic coupling and magnetic phase diagrams of the buckled-kagomé antiferromagnet \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Cu}_3\hbox {Bi}{(\hbox {SeO}_3)_2}\hbox {O}_2\hbox {Cl}$$\end{document}Cu3Bi(SeO3)2O2Cl

Single crystals of Cu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document}3Bi(SeO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document}3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2Cl were investigated using high-resolution capacitance dilatometry in magnetic fields up to 15 T. Pronounced magnetoelastic coupling is found upon evolution of long-range antiferromagnetic order at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm {N}}$$\end{document}TN \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$= 26.4(3)$$\end{document}=26.4(3) K. Grüneisen analysis reveals moderate effects of uniaxial pressure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm {N}}$$\end{document}TN, of 1.8(4) K/GPa, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-0.62(15)$$\end{document}-0.62(15) K/GPa and 0.33(10) K/GPa for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \parallel a$$\end{document}p‖a, b, and c, respectively. Below 22 K Grüneisen scaling fails which implies the presence of competing interactions. The structural phase transition at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm {S}}$$\end{document}TS \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$= 120.7(5)$$\end{document}=120.7(5) K is much more sensitive to uniaxial pressure than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm {N}}$$\end{document}TN, with strong effects of up to 27(3) K/GPa (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \parallel c$$\end{document}p‖c). Magnetostriction and magnetization measurements reveal a linear magnetoelastic coupling for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\parallel c$$\end{document}B‖c below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm {N}}$$\end{document}TN, as well as a mixed phase behavior above the tricritical point around 0.4 T. An analysis of the critical behavior in zero-field points to three-dimensional (3D) Ising-like magnetic ordering. In addition, the magnetic phase diagrams for fields along the main crystalline axes are reported.


II. LATTICE AND COUPLINGS
The lattice of the francisite Cu 3 Bi(SeO 3 ) 2 O 2 Cl according to Ref  The volume thermal expansion, gained from summing over the a, b, and c axis, is shown in Fig. S3. A strong kink is seen in dV /V at T N ( Fig. S3(b)), corresponding to a positive λ-like anomaly in the volume expansion coefficient β ( Fig. S3(a)). At T S a jump in β occurs. The additional peak is most likely an artifact from summing the three linear thermal expansion coefficients of the a, b, and c axis to obtain the volume expansion.

B. Phononic Background Fits
Phononic background fits to the thermal expansion coefficients and the specific heat are shown in Fig. S4. Both thermal expansion and specific heat are fitted by Debye and Einstein contributions according to where n D1,2 and n E are constants, and D(T /Θ D1,2 ) and E(T /Θ E ) are the Debye and Einstein functions with the Debye and Einstein temperatures Θ D1,2 and Θ E . A fit to the specific heat ( Fig. S4(a)) below the onset of T S , at 35 K to 93 K, yields n D1 = 3.25, n D2 = 6.14 and n E = 12.1 with Θ D1 = 127 K, Θ D2 = 377 K and Θ E = 1063 K. These Debye and Einstein temperatures are then used to fit the thermal expansion data in the range from 35 K to 60 K. Due to the high value of the Einstein temperature, however, the contribution of Einstein modes to the thermal expansion below 100 K is negligible. Therefore, and in order to reduce the number of free parameters, the Einstein mode is omitted for the thermal expansion fit. The resulting background fits are shown in Fig. S4(b). The fits for the c axis and volume describe the data very well -the c axis up to about 80 K and the volume up to T S -while the a and b axis only coincide with the data in a narrow temperature range, roughly between 40 K and 54 K.

C. Extraction of Jump Heights
The extraction of jump heights at T S by an area-conserving (and entropy-conserving for c p ) method for the thermal expansion and specific heat is shown in Fig. S5. Jump heights are indicated in the figure.

E. Thermal Expansion at 0 T and 15 T
A comparison of thermal expansion data at 0 T and 15 T, normalized above T S , shows in-plane field effects up to about 100 K as described in the main article. (Fig. S7).

IV. FAILURE OF GRÜNEISEN SCALING BELOW TN
In the main text we mentioned the failure of Grüneisen scaling below 22 K (see Fig. 2). Here, we want to spend some more time to explore this behavior. Fig. S8 shows the magnetic contributions to the specific heat and the thermal expansion in zero-field and at B > 0 up to 7 T and 15 T, respectively. The same phononic background as described in the main text was used for all data sets. For better comparability with c p,mag /T we plotted α c,mag /T in Fig. S8(b). The specific heat data shows two anomalies, one at 15 K which is not affected my the magnetic field and one which shifts to higher temperatures and broadens as the field is increased. The latter marks the crossover from the field-induced ferrimagnetic phase to the paramagnetic phase. Similarly, α c,mag /T shows a peak at 15 K which is insensitive to a magnetic field and an additional shoulder may be seen around 30 K at 1 T, which is indistinguishable from the high-temperature tail of the 15 K peak at 15 T. So what is the origin of the peak at 15 K? A Schottky anomaly can be ruled out, because in contrast to the observed behavior it would shift in an applied magnetic field unless it were from transitions of electrons between energy levels of the same spin quantum number m S . Plotting α c,mag /T 2 vs. T (Fig. S8(b) inset) shows a linear rise up to about 10 K, i.e., α c,mag ∝ T 3 , with a negative offset at B > 0. While the offset can be related to ferromagnetic magnons -α FM ∝ c p,FM ∝ T 3/2 -the linear behavior signals either phononic or antiferromagnetic (AFM) magnonic contributions. In the ferromagnetic phase above the critical field no AFM magnons can be present, therefore the linear rise suggests a phononic origin of the 15 K anomaly judging from the thermal expansion data. In contrast, the rise of c p,mag /T 2 (inset of Fig. S8(a)) is not as clearly linear and additional effects which do not show up in the thermal expansion may be present, leading to the failure of Grüneisen scaling. Furthermore, a global spin gap of 1.57 meV (18.2 K) was reported from inelastic neutron scattering experiments. 4 This fits with the temperature scale of the anomaly and would suggest a relation to magnonic excitations. The presence of the anomaly in zero-field, i.e., in the AFM phase, would then be explained by ferromagnetic spin-waves within the ferromagnetically coupled layers.
Lastly, two resonances at 1.23 meV (14.3 K) and 1.28 meV (14.8 K) were observed in the brother compound Cu 3 Bi(SeO 3 ) 2 O 2 Br in time-domain THz spectra at 3.9 K. 5 Electron spin resonance (ESR) measurements from the same report suggest that these two resonances are of magnetic origin and shift to higher frequencies in higher magnetic fields. However, a flat, i.e., field-independent, resonance would not be seen by spectroscopic field-sweeps around 300 GHz. Frequency sweeps at different magnetic fields would be necessary to observe such a resonance.
In conclusion, low-energy optical phonons, potentially coupled to magnon excitations, seem to be the cause for the anomalies observed around 15 K.

V. MAGNETIZATION MEASUREMENTS
The isothermal magnetization at 2 K up to 7 T as well as the static magnetic susceptibility at 1 T are shown in Fig. S9. Measurements of the isothermal magnetization up to 14 T at temperatures up to 25 K (a and b axis) and up to 50 K (c axis) are shown in Fig. S10. Phase boundaries for the phase diagrams (Fig. 6) were obtained from the peaks in the magnetic susceptibility ∂M i /∂B. The static magnetic susceptibility for all axes between 1 T and 14 T is shown together with the Fisher specific heat in Fig. S11. Phase boundaries for the phase diagrams in Fig. 6 were obtained from the peaks in the Fisher specific heat for the in-plane directions and the c axis up to 0.5 T. Above 0.5 T phase boundaries were obtained from the temperature at half of the jump heights upon entering and exiting the intermediate mixed phase. Table S1 shows the experimental results from our magnetization measurements in comparison to the results of calculations by Nikolaev et al.     Table S2 shows the quantities extracted from magnetization and dilatometry measurements upon sweeping the magnetic field (B) or the temperature (T ) and the resulting calculated quantities obtained according to Eq. (2) to (4) in the main text.

VII. CRITICAL SCALING ANALYSIS
The magnetic contributions to the thermal expansion coefficients and the specific heat in Fig. 7 in the main text were fitted by Eq. (5), i.e., where t = T /T N −1 is the reduced temperature. Initially, this left us with five free fit parameters for each fit. However, since the phononic contributions to c p and α i were already subtracted, both the offset B and the linear term D ± were set to zero, leaving us with only three fit parameters. To further reduce the free parameters to two, we tried fixing α ± , either to values from fits for T < T N or to a value giving good fit results over a wide range, also beyond the actual fitting range. Our best least-square fitting results are presented in Tab. S3. For comparison results with fixed and free values for α ± are shown for α a,mag and c p,mag . Table S3: Fit parameters for the critical scaling according to Eq. (5) in the main text. Unless indicated otherwise, B = 0 and D = 0 were fixed for all fits. The parameters A ± are in units of 1/K and J/(mol K), for thermal expansion and specific heat, respectively. 'Range' gives the fitting range. '(f)' indicates that the quantity was fixed manually and not fitted.